Angular Velocity: Sect. 1.15 Overview only. For details, see text!

• Consider a particle moving on arbitrary path in space: – At a given instant, it can be considered as moving in a plane,

circular path about an axis

Instantaneous Rotation Axis. In an infinitesimal time dt, the path can be represented as infinitesimal circular arc.

• As the particle moves in circular path, it has angular velocity:

ω (dθ/dt) θ

• Consider a particle moving in an instantaneously circular path of radius R. (See Fig.): – Magnitude of Particle

Angular Velocity:

ω (dθ/dt) θ– Magnitude of

Linear Velocity

(linear speed):

v = R(dθ/dt)

= Rθ = Rω

• Particle moving in circular path, radius R. (Fig.):

Angular Velocity: ω θ

Linear Speed: v = Rω (1)

• Vector direction of ω

normal to the plane of motion,

in the direction of a right hand

screw. (Fig.). Clearly:

R = r sin(α) (2)

(1) & (2) v = rωsin(α)

So (for detailed proof, see text!):

v = ω r

Gradient (Del) Operator: Sect. 1.16 Overview only. For details, see text!

• The most important vector differential operator: grad

A Vector which has components which are differential operators. Gradient operator.

• In Cartesian (rectangular) coordinates:

∑i ei (∂/∂xi) (1)

• NOTE! (For future use!) is much more complicated in cylindrical & spherical coordinates (see Appendix F)!!

can operate directly on a scalar function

( gradient of Old Notation: = grad):

= ∑i ei (∂/∂xi) A VECTOR!

can operate in a scalar product with a vector A ( divergence of A; Old: A = div A):

A = ∑i (∂Ai/∂xi) A SCALAR!

can operate in a vector product with a vector A ( curl of A; Old: A = curl A):

(A)i = ∑j,k εijk(∂Ak/∂xj) A VECTOR!

(Older: A = rot A) Obviously, A = A(x,y,z)

• Physical interpretation of the gradient : (Fig)

• The text shows that has the properties:

1. It is surfaces of constant 2. It is in the direction of max change in 3. The directional derivative of for any direction n is n = (∂/∂n)

(x,y)

Contour plot of (x,y)

The Laplacian Operator• The Laplacian is the dot product of with itself:

2 ; 2 ∑i (∂2/∂xi2)

A SCALAR!

• The Laplacian of a scalar function 2 ∑i (∂2/∂xi

2)

Integration of Vectors: Sect. 1.17 Overview only. For details, see text!

• Types of integrals of vector functions:

A = A(x,y,z) = A(x1,x2,x3) = (A1,A2,A3)

• Volume Integral (volume V, differential volume element dv) (Fig.):

∫V A dv (∫V A1dv, ∫V A2dv, ∫V A3dv)

• Surface Integral (surface S, differential surface element da) (Fig.)

∫S An da, n Normal to surface S

• Line integral (path in space, differential path element ds) (Fig.):

∫BC Ads ∫BC ∑i Ai dxi

• Gauss’s Theorem or Divergence Theorem (for a closed surface S surrounding a volume V)

See figure; n Normal to surface S

∫S An da = ∫V A dv

Physical Interpretation of AThe net “amount” of A “flowing” in & out of closed surface S

• Stoke’s Theorem (for a closed loop C surrounding a surface S) See Figure; n Normal to surface S

∫C Ads = ∫S (A)n da

Physical Interpretation of AThe net “amount” of “rotation” of A